Rehman, Abdur; Wang, Qing-Wen; Ali, Ilyas; Akram, Muhammad; Ahmad, M. O. A constraint system of generalized Sylvester quaternion matrix equations. (English) Zbl 1386.15035 Adv. Appl. Clifford Algebr. 27, No. 4, 3183-3196 (2017). Summary: By keeping in mind the great number of applications of generalized Sylvester matrix equations in systems and control theory, in this paper we establish some necessary and sufficient conditions for the solvability to a system of eight generalized Sylvester matrix equations over the quaternion algebra. The general solution to this system is also constructed when it is solvable. Moreover, an algorithm and a numerical example are also given to make the results of this paper more practical in various fields of engineering. The findings of this paper generalize previous results in the literature. Cited in 17 Documents MSC: 15A24 Matrix equations and identities 15A03 Vector spaces, linear dependence, rank, lineability 15A09 Theory of matrix inversion and generalized inverses 15B33 Matrices over special rings (quaternions, finite fields, etc.) 15B57 Hermitian, skew-Hermitian, and related matrices 11R52 Quaternion and other division algebras: arithmetic, zeta functions Keywords:quaternion; generalized Sylvester matrix equation; exclusive solution; Moore-Penrose inverse; rank PDFBibTeX XMLCite \textit{A. Rehman} et al., Adv. Appl. Clifford Algebr. 27, No. 4, 3183--3196 (2017; Zbl 1386.15035) Full Text: DOI References: [1] Adler, S.L.: Quaternionic Quantum Mechanics and Quantum Fields. Oxford University Press, New York (1995) · Zbl 0885.00019 [2] Bai, Z.Z.: On Hermitian and skew-Hermitian splitting iteration methods for continuous Sylvester equations. J. Comput. Math. 29, 185-198 (2011) · Zbl 1249.65090 [3] Baksalary, J.K., Kala, R.: The matrix equation \[AX-YB=C\] AX-YB=C. Linear Algebra Appl. 25, 41-43 (1979) · Zbl 0403.15010 [4] Barraud, A., Lesecq, S., Christov, N.: From sensitivity analysis to random floating point arithmetics-application to Sylvester equations. Numer. Anal. Appl. Lect. Notes Comput. Sci. 2001, 35-41 (1988) · Zbl 0978.65031 [5] Buxton, J.N., Churchouse, R.F., Tayler, A.B.: Matrices Methods and Applications. Clarendon Press, Oxford (1990) [6] Darouach, M.: Solution to Sylvester equation associated to linear descriptor systems. Syst. Control Lett. 55, 835-838 (2006) · Zbl 1100.93028 [7] Dmytryshyn, A., Kåström, B.: Coupled Sylvester-type matrix equations and block diagonalization. SIAM J. Matrix Anal. Appl. 38, 580-593 (2015) · Zbl 1328.15028 [8] Farid, F.O., He, Z.H., Wang, Q.W.: The consistency and the exact solutions to a system of matrix equations. Linear Multilinear Algebra 64(11), 2133-2158 (2016) · Zbl 1357.15007 [9] Gavin, K.R., Bhattacharyya, S.P.: Robust and well-conditioned eigenstructure assignment via Sylvester’s equation. In: Proceedings of American Control Conference (1982) [10] Hamilton, W.R.: On quaternions; or on a new system of imaginaries in algebra. Philos. Mag. 25(3), 489-495 (1844) [11] He, Z.H., Wang, Q.W.: A System of periodic discrete-time coupled Sylvester quaternion matrix equations. Algebra Coll. 24(1), 169-180 (2017) · Zbl 1361.15015 [12] He, Z.H., Wang, Q.W.: The general solutions to some systems of matrix equations. Linear Multilinear Algebra 63(10), 2017-2032 (2015) · Zbl 1334.15040 [13] He, Z.H., Wang, Q.W.: A pair of mixed generalized Sylvester matrix equations. J. Shanghai Univ. Nat. Sci. 20, 138-156 (2014) · Zbl 1313.15029 [14] Kågström, B.: A perturbation analysis of the generalized Sylvester equation \[(AR - LB, DR - LE) = (C, F )\](AR-LB,DR-LE)=(C,F). SIAM J. Matrix Anal. Appl. 15, 1045-1060 (1994) · Zbl 0805.65045 [15] Kyrchei, I.I.: Determinantal representation of the Moore-Penrose inverse matrix over the quaternion skew field. J. Math. Sci. 180(1), 23-33 (2012) [16] Kyrchei, I.I.: Explicit representation formulas for the minimum norm least squares solutions of some quaternion matrix equations. Linear Algebra Appl. 438(1), 136-152 (2013) · Zbl 1255.15022 [17] Kyrchei, I.I.: Weighted singular value decomposition and determinantal representations of the quaternion weighted Moore-Penrose inverse. Appl. Math. Comput. 309, 1-16 (2017) · Zbl 1411.15025 [18] Leo, S.D., Scolarici, G.: Right eigenvalue equation in quaternionic quantum mechanics. J. Phys. A. 33, 2971-2995 (2000) · Zbl 0954.81008 [19] Lee, S.G., Vu, Q.P.: Simultaneous solutions of matrix equations and simultaneous equivalence of matrices. Linear Algebra Appl. 437, 2325-2339 (2012) · Zbl 1253.15024 [20] Li, R.C.: A bound on the solution to a structured Sylvester equation with an application to relative perturbation theory. SIAM J. Matrix Anal. Appl. 21(2), 440-445 (1999) · Zbl 0964.15018 [21] Lin, Y.Q., Wei, Y.M.: Condition numbers of the generalized Sylvester equation. IEEE Trans. Autom. Control 52, 2380-2385 (2007) · Zbl 1367.65069 [22] Marsaglia, G., Styan, G.P.H.: Equalities and inequalities for ranks of matrices. Linear Multilinear Algebra 2, 269-292 (1974) · Zbl 0297.15003 [23] Rehman, A., Wang, Q.W.: A system of matrix equations with five variables. Appl. Math. Comput. 271, 805-819 (2015) · Zbl 1410.15033 [24] Rehman, A., Wang, Q.W., He, Z.H.: Solution to a system of real quaternion matrix equations encompassing \[\eta\] η-Hermicity. Appl. Math. Comput. 265, 945-957 (2015) · Zbl 1410.15034 [25] Roth, W.E.: The equations \[AX-YB=C\] AX-YB=C and \[AX-XB=C\] AX-XB=C in matrices. Proc. Am. Math. Soc. 3, 392-396 (1952) · Zbl 0047.01901 [26] Syrmos, V.L., Lewis, F.L.: Output feedback eigenstructure assignment using two Sylvester equations. IEEE Trans. Autom. Control 38, 495-499 (1993) · Zbl 0791.93035 [27] Syrmos, V.L., Lewis, F.L.: Coupled and constrained Sylvester equations in system design. Circ. Syst. Signal Process. 13(6), 663-694 (1994) · Zbl 0864.93050 [28] Took, C.C., Mandic, D.P.: Augmented second-order statistics of quaternion random signals. Signal Process. 91, 214-224 (2011) · Zbl 1203.94057 [29] Wang, Q.W., Rehman, A., He, Z.H., Zhang, Y.: Constraint generalized Sylvester matrix equations. Automatica 69, 60-64 (2016) · Zbl 1338.93174 [30] Wang, Q.W., He, Z.H.: Systems of coupled generalized Sylvester matrix equations. Automatica 50, 2840-2844 (2014) · Zbl 1300.93087 [31] Wang, L., Wang, Q.W., He, Z.H.: The common solution of some matrix equations. Algebra Coll. 23(1), 71-81 (2016) · Zbl 1348.15012 [32] Wang, Q.W., He, Z.H.: Solvability conditions and general solution for the mixed Sylvester equations. Automatica 49, 2713-2719 (2013) · Zbl 1364.15011 [33] Wang, Q.W., Wu, Z.C., Lin, C.Y.: Extremal ranks of a quaternion matrix expression subject to consistent systems of quaternion matrix equations with applications. Appl. Math. Comput. 182, 1755-1764 (2006) · Zbl 1108.15014 [34] Wimmer, H.K.: Consistency of a pair of generalized Sylvester equations. IEEE Trans. Autom. Control 39, 1014-1016 (1994) · Zbl 0807.93011 [35] Zhang, X.: A system of generalized Sylvester quaternion matrix equations and its applications. Appl. Math. Comput. 273, 74-81 (2016) · Zbl 1410.15037 [36] Zhang, Y.N., Jiang, D.C., Wang, J.: A recurrent neural network for solving Sylvester equation with time-varying coefficients. IEEE Trans. Neural Netw. 13(5), 1053-1063 (2002) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.